Plane model-fields of definition, fields of definition, and the field of moduli for smooth plane curves

摘要

Let C/(k) over bar be a smooth plane curve defined over (k) over bar, a fixed algebraic closure of a perfect field k. We call a subfield k' subset of (k) over bar a plane model field of definition for C if C descends to k' as a smooth plane curve over k', that is if there exists a smooth curve C'/k' defined over k' which is k'-isomorphic to a non-singular plane model F(X,Y, Z) = 0 with coefficients in k', and such that C' circle times(k') and C are isomorphic. In this paper, we provide (explicit) families of smooth plane curves for which the three fields types; the field of moduli, fields of definition, and plane-models fields of definition are pairwise different. (C) 2018 Elsevier Inc. All rights reserved.